The global real interest rate is arguably the most important price in world financial markets. It determines the rate at which agents in the world economy are willing to substitute today’s consumption for consumption in the future, and, at the same time, it signals the pressures that investment demands impose on the supply of world saving. By some accounts, the global real interest rate in the mid-1990s is high by historical standards. Is it? Do world economic policies influence the real interest rate over the medium to long term, or do other factors, such as technological change, matter more? And has government dissaving really been a driving force behind the rise of the global real interest rate since the early 1980s, as claimed by some? This paper analyzes these issues. It has two main objectives. First, it develops measures of the global real interest rate and provides historical evidence to help judge whether the rate today is high or low. Second, it examines the factors that influence the global real interest rate over time and quantifies the extent to which shifts in government policies and other factors have caused it to move.

A key challenge in economic research has been to estimate the impact of government dissaving on real interest rates. Many economists believe that there must be some linkage because of crowding-out effects, but most studies over the past decade have found little empirical support for this hypothesis. Blanchard and Summers (1984) provide one of the first examinations of the increase in global real interest rates in the 1980s. They conclude that most of the increase was due to higher stock market returns, not bigger budget deficits. Barro and Sala-i-Martin (1990) and Barro (1992) use an analytical framework in which saving equals investment to study the determinants of short-term real interest rates, and they also find little effect of fiscal policy. More recently, Gagnon (1995) claims that, if ten-year historical inflation is a good proxy for inflation expectations, real U.S. interest rates did not really increase much in the 1980s and 1990s, and so there is nothing for the larger budget deficits or higher ratios of government debt to GDP to explain.

This paper presents evidence that both short- and long-term global real interest rates have been higher since 1981 than they were in earlier periods. This finding is shown to be robust with respect to the inflation model and also with respect to different real interest rate concepts. The paper also finds that higher government debt in the world economy is associated with higher global real interest rates. In this sense, this study is more in agreement with the work of Howe and Pigott (1991) and Ford and Laxton (1995), who also find significant fiscal effects, than with the economists mentioned above.

A particular feature of this paper is that its analytical framework views the global real interest rate as determined through the equilibration of the supply and demand for world assets. It examines the relationship between a comprehensive global real interest rate and measures of the world rate of return on productive capacity, world government debt, and similar factors—not just national measures, as used in most studies—because, as world capital mobility steadily increases, these issues must be analyzed globally.

This paper has two main pans. The first part looks at the concept of the global interest rate, describes the construction of various global interest rate measures, and tests for the existence of different interest rate regimes. The second part develops a framework for modeling the global interest rate and provides empirical evidence about the factors that seem to explain its movements.

## Measurement of the Global Real Interest Rate

In a world with integrated capital markets, arbitrage ensures that (risk-adjusted) returns on similar assets are equalized across countries.^{1} Real interest rates are among the most important rates of return, for they determine the rate at which economic agents are willing to substitute present for future consumption.^{2} Saving is the reflection of this decision to forgo consumption today; at the same time, it is also related to the rate of capital accumulation and growth in the world economy. It seems natural, therefore, to include real interest rates in the set of indicators used to evaluate the current state of the world economy. Using a single real interest rate as a sufficient statistic—in the sense that it contains (almost) all the relevant information contained in national real interest rates—could greatly simplify the task. A single real interest rate, which is an average rate at which agents in the world economy are willing to substitute consumption today for consumption in the future, is, therefore, what is meant by the term “global real interest rate.”

Barro and Sala-i-Martin (1990) construct a global real interest rate that equilibrates saving and investment in the world economy. Their theoretical basis is the proposition that there must be a common component in real interest rates in all countries that are integrated in the world capital market. If, as is claimed in their paper, this common component is one of the most important sources of variation in real interest rate movements over time, the global real interest rate is a useful concept.

The common component in national real interest rates is the result of transactions in asset and goods markets, both of which are channels of transmission of shocks between countries. In Appendix I, it is shown that in a very stylized model the common component can be thought of as the interest rate for a synthetic asset, a real bond denominated in terms of world output. It is, therefore, an average price for a fictitious world consumer.

The global real interest rate is essentially an empirical concept, although it can be derived rigorously as an implied rate of return in the context of standard asset-pricing relations (as in Appendix I, for example). The reason is that the common component concept puts fewer restrictions on the arbitrage process and expectation formation than most standard asset-pricing models. If assets are priced according to some model, the global real interest rate follows as an implied price. It can still be calculated, however, even if most asset prices cannot be explained by standard models, as long as national real interest rates are correlated.^{3} The strong empirical bias is both an advantage and a disadvantage. The concept benefits because it does not require strong assumptions that do not hold on empirical grounds, but it suffers from the lack of theoretical implications that could be tested.

## Review of the Measurement of the Global Real Interest Rate

Various methods to extract common factors from a set of national consumption-based real interest rates have been suggested in the literature. Conceptually, it is possible to distinguish between methods that derive the global real interest rate directly and methods that derive a rate by restricting the deviations of national real interest rates from the global real interest rate. The first class of methods assume a common component, whereas, in the second method, the common component is estimated.

Barro and Sala-i-Martin (1990) define the short-term world real interest rate as a weighted average of *k* national short-term real interest rates:

where *r* denotes an expected real interest rate and *k* countries, assuming that there is one representative investor per country. These investors care about the return in terms of their consumption basket. It is also assumed that there is one bond for each possible maturity and currency. In a setup with *K* countries, this average is given at any time *t* by

The first summation averages all the outstanding bonds of the same maturity in the world. The second summation measures the real return on the bond in currency *k* over all investors *j*. *k* at time t, *k* and the reference currency *K* at time (*(X ^{K}* = 0), and

*j*(

*j*and the reference currency, and

*t*and t+1 in country

*j*).

*k*by investors in country

*j*, and

The implications in terms of restrictions are easier to understand if equation (2) is rewritten for the case of two countries, two currencies, and two noncontingent assets that are free of default risk:

Barro and Sala-i-Martin’s weighted average is a good approximation if (a) GDP weights approximate the relative asset supply well, (b) cross-border holdings are close to zero, or (c) all terms except the first two are independently and identically distributed random variables at any point in time (except, possibly, for a constant mean deviation related to the country size).

The weighted-average method can also be considered to be a generalization of the standard consumption formula for pricing capital assets for the risk-free real interest rate, which states that the real interest rate is simply the sum of the rate-of-time preference and the average growth rate of consumption times the intertemporal elasticity of substitution (and a risk term).^{4} Any national real interest rate is then determined by the growth rate of national consumption (and the other terms), so that a weighted average of national interest rates is an implicit real interest rate consisting of an average rate-of-time preference and the growth rate of world GDP (weighted by the intertemporal elasticity of substitution). Deviations from the world real interest rate should also be independently and identically distributed over time for the weighted-average method to be appropriate.

Gagnon and Unferth (1993) use a panel data model to estimate global real interest rates. Each national real interest rate is explained by a common world component, which is modeled by a constant and a time factor, as well as by a country-specific factor, which is modeled by a time-invariant dummy. Using annual data, they conclude that interest differentials disappear within a year, that capital markets are integrated, and that a global real interest rate, defined as a common factor, does exist. Their approach imposes more structure on the global real interest rate than the weighted-average method. The world real interest rate is estimated by imposing a model for its mean and the residuals, which have to be independently and identically distributed over time. Thus, Gagnon and Unferth’s model implies that, at any point in time, the world real interest rate is a simple average of national real interest rates; however, Gagnon and Unferth provide little economic reasoning for this model of the world real interest rate.

Ford and Laxton (1995) use the first principal component of a set of national real interest rates to construct a proxy for the global real interest rate. The principal component approach is appealing because it can be regarded as a kind of a factor asset pricing model. Excess returns on noncontingent assets that are free of default risk, defined as deviations of national interest rates from the global real interest rate, can be modeled as a function of one or more factors. The problem with this approach is that the risk-free rate of return or, in this case, the world real interest rate, is not defined. Ford and Laxton standardize their raw input series, a set of national real interest rates, to obtain a unique factor representation. The global real interest rate is constructed by renormalizing the squared load factors implied by the first principal component, such that they sum to unity, and then using them as weights for the nonstandardized national real interest rates. However, this approach might bias the level of the global real interest rate because it is constructed with weights derived from deviations.

Brunner and Kaminsky (1994) consider the global real interest rate to be the common trend driving cointegrated national real interest rates. If deviations from bilateral real interest parity are stationary for all possible combinations of a set of national real interest rates, there can be only one common trend, that is, a common nonstationary component. The common trend approach shares the idea underlying the principal component methodology. It has the advantage, however, that it can allow for shifts in the mean. The common trend, a random walk, represents simply a mean shift in every period. This concept is too extreme to be literally true, but it might be a good statistical representation of real interest rates that display unit root properties over fairly long samples. The common trend also can be expressed as a weighted average of national real interest rates in the form

where the weights are constant over time and, in general, are not equal to GDP weights, (See Appendix III for a more detailed discussion of how the weights can be determined.)

None of these methods is unambiguously superior. The weighted-average approach is pragmatic and offers intuitive appeal, including the feature that the size of a country’s economy and its asset markets matter. It also produces a global real interest rate whose mean value is financially meaningful, which allows for an analytical connection to rates of return on stock markets and other real world measures. The assumption that deviations owing to expected real exchange rate changes are independently distributed over time is rather strong, however. Similar arguments apply to the simple averages implied by the panel data approach. The principal component method is appealing because it uses information contained in the data, but it suffers from using standardized measures, which leave mean values unexplained. Because no single approach is clearly best, three methods are tested in this paper.

### Constructing the Global Real Interest Rate

Data for Belgium, Canada, Germany, Japan, the Netherlands, Switzerland, the United Kingdom, and the United States were used to construct global real interest rates according to three of these methods-—the weighted average, principal component, and common trend methodology.^{5} The eight countries were selected because they had relatively open capital markets for the period from 1960 to 1994, and because they offer historical data back to the 1960s. The national short-term real interest rates are ex post real interest rates on a quarterly basis, constructed by subtracting the one-period-ahead annualized three-month inflation rate, as measured by the consumer price index, from a representative three-month interest rate.^{6} (Summary statistics can be found in Appendix III, Table A1.) Long-term ex ante national real interest rates require an inflation forecast, and an exponential-smoothing formula was used to update the expected annual inflation rate based on consumer prices (see Appendix II).^{7} The smoothed value was then subtracted from a representative quarterly long-term government bond yield. Other standard univariate methods of modeling inflation expectations were also developed, but the resulting real long-term interest rates did not differ significantly.

Two sets of global real interest rates were generated for both short- and long-term real interest rates with all three methods (Table 1). The first set includes all eight countries in the sample, while the second set includes just the United States. Japan, and Germany—the so-called G-3 countries. For the rates computed with the weighted-average method, the weights in any time period are given by the share of each country’s nominal GDP (in U.S. dollars at current market exchange rates) in the total GDP of all countries included in the calculations in the same period. GDP in current market prices, as well as market exchange rates, is likely to reflect the supply of assets more accurately than GDP weights based on purchasing power parity exchange rates because market clearing in financial markets occurs in nominal terms.^{8}

**Summary Statistics for Global Short- and Long-Term Real Interest Rates**

^{1}In percent; standard deviation in parentheses.

^{2}*t*-statistics of the coefficient on the lagged level of series in an augmented Dickey-Fuller unit root regression. Lag length (in parentheses) selected according to significance of coefficients of lagged first difference of series (Campbell and Perron (1991)).

^{3}Minimum t-statistics of the coefficient on the lagged level of a series of augmented Dickey-Fuller unit root regressions with a 1-0 dummy variable for every period in the sample (excluding end points). Lag length in parentheses. Critical values for lest statistics taken from Perron and Vogelsang (1992).

^{4}Period in which minimum t-statistic occurs.

^{5}Belgium. Canada, Germany, Japan, Netherlands, Switzerland. United Kingdom, and United States.

^{6}Germany. Japan, and the United States.

^{7}See Appendix III for details.

**Summary Statistics for Global Short- and Long-Term Real Interest Rates**

Mean^{1} | Break | ||||||||
---|---|---|---|---|---|---|---|---|---|

60:QI-94:QIII | 60:QI-72:QIV | 73lQI-80:QIV | 81:QI-94:QIII | ADF t-statistics^{2} | ADF t-statistics^{3} | Date^{4} | |||

Short-term real interest rates | |||||||||

Weighted average | |||||||||

Eight countries^{5} | 1.631 | 1.169 | -1.230 | 3.733 | -2.026 (6) | -4.015 (6) | 79:QIV | ||

(2.568) | (1.140) | (2.309) | (1.734) | ||||||

Three countries^{6} | 1.636 | 1.229 | -1.081 | 3.603 | -2.210 (2) | -4.069 (6) | 79:QIV | ||

(2.556) | (1.100) | (2.417) | (1.929) | ||||||

Principal component^{7} | |||||||||

Eight countries^{5} | 1.939 | 0.945 | -0.773 | 4.457 | -1.526 (4) | -3.498 (6) | 79:QII | ||

(2.846) | (1-917) | (2.499) | (1.372) | ||||||

Three countries^{6} | 1.734 | 1.167 | -0.730 | 3.703 | -2.230 (3) | -4.407 (6) | 80:QI | ||

(2.812) | (1.945) | (3.187) | (1.716) | ||||||

Common trend^{7} | |||||||||

Eight countries^{5} | 1.769 | 0.718 | -0.847 | 4.286 | -1.557 (4) | -3.588 (6) | 79:QII | ||

(2.948) | (2.135) | (2.719) | (1.446) | ||||||

Three countries^{6} | 1.952 | 1.311 | -0.212 | 3.818 | 2.241 (3) | -4.578 (6) | 80:QI | ||

(2.625) | (2.008) | (2.522) | (1.585) | ||||||

Long-term real interest rates | |||||||||

Weighted average | |||||||||

Eight countries^{5} | 3.084 | 2.978 | 0.451 | 4.716 | -2.137 (3) | -4.054 (3) | 80:QII | ||

(1.944) | (0.441) | (1.457) | (1.193) | ||||||

Three countries^{6} | 3.235 | 3.058 | 0.586 | 4.945 | -1.927 (4) | -4.331 (3) | 80:QII | ||

(2.003) | (0.421) | (1.475) | (1.306) | ||||||

Principal component^{7} | |||||||||

Eight countries^{5} | 3.037 | 2.927 | 0.433 | 4.655 | -1.443 (6) | -3.019 (3) | 83:QI | ||

(1.956) | (0.816) | (1.902) | (0.821) | ||||||

Three countries^{6} | 3.247 | 3.557 | 0.377 | 4.624 | -1.808 (4) | -3.375 (4) | 80:QIII | ||

(2.117) | (1.150) | (1.971) | (0.990) | ||||||

Common trend^{7} | |||||||||

Eight countries^{5} | 3.250 | 3.079 | 0.757 | 4.864 | -1.216 (5) | -3.217 (3) | 80:QIII | ||

(1.903) | (0.704) | (1.747) | (0.817) | ||||||

Three countries^{6} | 3.439 | 3.584 | 0.978 | 4.734 | -1.749 (4) | -3.650 (3) | 80:QIII | ||

(1.775) | (0.819) | (1.415) | (0.976) |

^{1}In percent; standard deviation in parentheses.

^{2}*t*-statistics of the coefficient on the lagged level of series in an augmented Dickey-Fuller unit root regression. Lag length (in parentheses) selected according to significance of coefficients of lagged first difference of series (Campbell and Perron (1991)).

^{3}Minimum t-statistics of the coefficient on the lagged level of a series of augmented Dickey-Fuller unit root regressions with a 1-0 dummy variable for every period in the sample (excluding end points). Lag length in parentheses. Critical values for lest statistics taken from Perron and Vogelsang (1992).

^{4}Period in which minimum t-statistic occurs.

^{5}Belgium. Canada, Germany, Japan, Netherlands, Switzerland. United Kingdom, and United States.

^{6}Germany. Japan, and the United States.

^{7}See Appendix III for details.

**Summary Statistics for Global Short- and Long-Term Real Interest Rates**

Mean^{1} | Break | ||||||||
---|---|---|---|---|---|---|---|---|---|

60:QI-94:QIII | 60:QI-72:QIV | 73lQI-80:QIV | 81:QI-94:QIII | ADF t-statistics^{2} | ADF t-statistics^{3} | Date^{4} | |||

Short-term real interest rates | |||||||||

Weighted average | |||||||||

Eight countries^{5} | 1.631 | 1.169 | -1.230 | 3.733 | -2.026 (6) | -4.015 (6) | 79:QIV | ||

(2.568) | (1.140) | (2.309) | (1.734) | ||||||

Three countries^{6} | 1.636 | 1.229 | -1.081 | 3.603 | -2.210 (2) | -4.069 (6) | 79:QIV | ||

(2.556) | (1.100) | (2.417) | (1.929) | ||||||

Principal component^{7} | |||||||||

Eight countries^{5} | 1.939 | 0.945 | -0.773 | 4.457 | -1.526 (4) | -3.498 (6) | 79:QII | ||

(2.846) | (1-917) | (2.499) | (1.372) | ||||||

Three countries^{6} | 1.734 | 1.167 | -0.730 | 3.703 | -2.230 (3) | -4.407 (6) | 80:QI | ||

(2.812) | (1.945) | (3.187) | (1.716) | ||||||

Common trend^{7} | |||||||||

Eight countries^{5} | 1.769 | 0.718 | -0.847 | 4.286 | -1.557 (4) | -3.588 (6) | 79:QII | ||

(2.948) | (2.135) | (2.719) | (1.446) | ||||||

Three countries^{6} | 1.952 | 1.311 | -0.212 | 3.818 | 2.241 (3) | -4.578 (6) | 80:QI | ||

(2.625) | (2.008) | (2.522) | (1.585) | ||||||

Long-term real interest rates | |||||||||

Weighted average | |||||||||

Eight countries^{5} | 3.084 | 2.978 | 0.451 | 4.716 | -2.137 (3) | -4.054 (3) | 80:QII | ||

(1.944) | (0.441) | (1.457) | (1.193) | ||||||

Three countries^{6} | 3.235 | 3.058 | 0.586 | 4.945 | -1.927 (4) | -4.331 (3) | 80:QII | ||

(2.003) | (0.421) | (1.475) | (1.306) | ||||||

Principal component^{7} | |||||||||

Eight countries^{5} | 3.037 | 2.927 | 0.433 | 4.655 | -1.443 (6) | -3.019 (3) | 83:QI | ||

(1.956) | (0.816) | (1.902) | (0.821) | ||||||

Three countries^{6} | 3.247 | 3.557 | 0.377 | 4.624 | -1.808 (4) | -3.375 (4) | 80:QIII | ||

(2.117) | (1.150) | (1.971) | (0.990) | ||||||

Common trend^{7} | |||||||||

Eight countries^{5} | 3.250 | 3.079 | 0.757 | 4.864 | -1.216 (5) | -3.217 (3) | 80:QIII | ||

(1.903) | (0.704) | (1.747) | (0.817) | ||||||

Three countries^{6} | 3.439 | 3.584 | 0.978 | 4.734 | -1.749 (4) | -3.650 (3) | 80:QIII | ||

(1.775) | (0.819) | (1.415) | (0.976) |

^{1}In percent; standard deviation in parentheses.

^{2}*t*-statistics of the coefficient on the lagged level of series in an augmented Dickey-Fuller unit root regression. Lag length (in parentheses) selected according to significance of coefficients of lagged first difference of series (Campbell and Perron (1991)).

^{3}Minimum t-statistics of the coefficient on the lagged level of a series of augmented Dickey-Fuller unit root regressions with a 1-0 dummy variable for every period in the sample (excluding end points). Lag length in parentheses. Critical values for lest statistics taken from Perron and Vogelsang (1992).

^{4}Period in which minimum t-statistic occurs.

^{5}Belgium. Canada, Germany, Japan, Netherlands, Switzerland. United Kingdom, and United States.

^{6}Germany. Japan, and the United States.

^{7}See Appendix III for details.

The first principal component of standardized short-and long-term real interest rates was calculated for the period from 1960:QI to 1994:QIII. Global real interest rates were then generated with weights based on the load factors of the first principal component. The estimation results are documented in Appendix III, Tables A2 and A3. The results of the common trend approach for eight countries have to be interpreted with caution because the estimation results suggest that the common trend representation might be inappropriate (see Appendix III, Tables A4 and A5).

## Establishing Upper and Lower Bounds for the Global Real Interest Rate

While no single optimal method exists for constructing a global real interest rate, a comparison among the rates generated by the various approaches allows for the establishment of upper and lower bounds for the global real interest rate. As seen in Table 1, the maximum difference in the mean of the global short-term real interest rates for the period from 1960:QI to 1994:QIII is 32 basis points. The largest difference between various mean estimates is for the period from 1973:QI to 1980:QIV. For the years since 1981, the maximum difference in the mean global short-term real interest is about 85 basis points. These differences can be largely attributed to the differences in the weights of Japanese and U.K. short-term real interest rates, as both countries exhibited large negative real interest rates in the 1970s. In the past 13 years, the variation can be attributed to differences in the U.S. weight. Both the principal component approach and the common trend approach generate much smaller weights for the United States and Japan than the weighted-average method. Nevertheless, the qualitative properties of the various estimates are not very different, as shown in Chart 1, and the long-term movements coincide to a large degree.

The differences in means and variances are even smaller for long-term real interest rates. The estimates differ at most by 42 basis points between 1960:QI and 1994:QIII. For the period from 1981:QI to 1994:QIII, this divergence declines to just 32 basis points. The similarity of the various long-term global real interest rates can also be seen in Chart 2. Theseresults suggest that the analysis can safely be continued with only one short-term and one long-term global real interest rate.^{9} The choice was made to work with global real interest rates constructed according to the weighted-average method because it generates mean estimates that can be rationalized most easily on the grounds of economic theory.

**Annual Long-Term World Real Interest Rates**

(In percent)

Source: IMF staff estimates.^{1}Results for each method based on full sample of eight countries.

^{2}Germany, Japan, and United States.

^{3}Belgium, Canada, Germany, Japan, Netherlands, Switzerland, United Kingdom, and United States.

**Annual Long-Term World Real Interest Rates**

(In percent)

Source: IMF staff estimates.^{1}Results for each method based on full sample of eight countries.

^{2}Germany, Japan, and United States.

^{3}Belgium, Canada, Germany, Japan, Netherlands, Switzerland, United Kingdom, and United States.

**Annual Long-Term World Real Interest Rates**

(In percent)

Source: IMF staff estimates.^{1}Results for each method based on full sample of eight countries.

^{2}Germany, Japan, and United States.

^{3}Belgium, Canada, Germany, Japan, Netherlands, Switzerland, United Kingdom, and United States.

## Have There Been Different Global Real Interest Rates Regimes over the Past Third of a Century?

A main reason for attempting to define and construct a global real interest rate is to answer the question of whether it is high or low today, relative to earlier periods. The arbitrarily selected subperiods in Table 1 suggest that there have been three distinct global interest rate regimes over the past third of a century—the first between 1960 and 1972, the second between 1973 and 1980 (the period of adjustment to the oil price shocks), and a third since 1981. In this subsection, the selection of dates for the different regimes is tested by means of a nonlinear time-series model, proposed by Hamilton (1989) and (1990), that incorporates regime changes. In this class of model, a random variable—the global real interest rate, in this case—is assumed to be normally distributed. The parameters of the distribution are not time invariant, however. They depend, at any point in time t, on the unobserved state of the world at time t, denoted with *S _{t}* which is itself a random variable and which in this model can take values from i = 1,…, 3. This implies the following model for the global real interest rate:

^{10}

The probability of switching between states in the subsequent period, that is, *Pr*{*s _{t}* =

*j*|

*s*-1 =

_{t}*i*} =

*p*, is given by the matrix

_{ij}The unconditional probability density function for the global real interest rate is a probability-weighted mixture of the densities for each regime. The probability weights are determined by the probability that the world is in state *i* at time *t*, given information up to time These probabilities can be evaluated using an adapted Kalman filter algorithm. The unconditional probability density function can then be used to construct the log likelihood function and to estimate the parameters in the usual way. The model was estimated using quarterly data for the period from 1960:QI to 1994:QIII for both short- and long-term global real interest rates.

Results can be found in Table 2 and Chart 3. The estimates for the mean global real interest rate for each regime differ less than the estimates reported in Table 1. These parameters also allow one to estimate the probability that a global real interest rate was in regime *i* in period *t*. Two different probability concepts must be distinguished. One is the conditional probability of *s _{t}=i*, given information up to time

*t*. These probabilities are plotted as black lines in Chart 3. The other concept is that of the smoothed probability that

*s*, given the information in the whole sample, that is from

_{t}= i*t = 1,…, T*. These smoothed probabilities, which are shown as blue lines in Chart 3, are interesting because they add information to the estimates of the unobserved regimes.

**Time-Series Model with Regime Changes for Global Real Interest Rates, 1960:QI-1994:QIII**

(Standard errors of parameter estimates in parentheses)

^{1}Weighted average.

**Time-Series Model with Regime Changes for Global Real Interest Rates, 1960:QI-1994:QIII**

(Standard errors of parameter estimates in parentheses)

Short-Term Global Real Interest Rate^{1} | Long-Term Global Real Interest Rate^{1} | |
---|---|---|

Mean | ||

μ_{1} (regime 1) | 1.432 | 2.822 |

(0.133) | (0.062) | |

μ_{2} (regime 2) | -1.525 | 0.437 |

(0.332) | (0.257) | |

μ_{3} (regime 3) | 3.986 | 4.653 |

(0.235) | (0.143) | |

Standard deviation | ||

σ_{1} (regime 1) | 0.892 | 0.139 |

(0.179) | (0.033) | |

σ_{2} (regime 2) | 3.298 | 2.002 |

(0.817) | (0.507) | |

σ_{3} (regime 3) | 2.687 | 1.193 |

(0.540) | (0.215) | |

Probability of switching | ||

p1 (between regime 1 and 2) | 0.989 | 0.973 |

(0.012) | (0.023) | |

p2 (between regime 2 and 3) | 0.965 | 0.964 |

(0.030) | (0.031) | |

p3 (between regime 3 and 1) | 0.975 | 0.990 |

(0.021) | (0.010) |

^{1}Weighted average.

**Time-Series Model with Regime Changes for Global Real Interest Rates, 1960:QI-1994:QIII**

(Standard errors of parameter estimates in parentheses)

Short-Term Global Real Interest Rate^{1} | Long-Term Global Real Interest Rate^{1} | |
---|---|---|

Mean | ||

μ_{1} (regime 1) | 1.432 | 2.822 |

(0.133) | (0.062) | |

μ_{2} (regime 2) | -1.525 | 0.437 |

(0.332) | (0.257) | |

μ_{3} (regime 3) | 3.986 | 4.653 |

(0.235) | (0.143) | |

Standard deviation | ||

σ_{1} (regime 1) | 0.892 | 0.139 |

(0.179) | (0.033) | |

σ_{2} (regime 2) | 3.298 | 2.002 |

(0.817) | (0.507) | |

σ_{3} (regime 3) | 2.687 | 1.193 |

(0.540) | (0.215) | |

Probability of switching | ||

p1 (between regime 1 and 2) | 0.989 | 0.973 |

(0.012) | (0.023) | |

p2 (between regime 2 and 3) | 0.965 | 0.964 |

(0.030) | (0.031) | |

p3 (between regime 3 and 1) | 0.975 | 0.990 |

(0.021) | (0.010) |

^{1}Weighted average.

**Probabilities of Regime Changes for Global Real Interest Rates**

(Based on GDP weightst)

Source: IMF staff estimates.**Probabilities of Regime Changes for Global Real Interest Rates**

(Based on GDP weightst)

Source: IMF staff estimates.**Probabilities of Regime Changes for Global Real Interest Rates**

(Based on GDP weightst)

Source: IMF staff estimates.This three-regime characterization, based upon sample statistics for subsample periods, seems to be appropriate. The degree of uncertainty about the current regime, as measured by the difference between the conditional and the smoothed probabilities, is surprisingly small, particularly in the case of the long-term global real interest rate. The estimated unconditional probabilities for the short- and long-term global real interest rates overlap almost perfectly.

## Determinants of the Global Real Interest Rate

### Previous Attempts to Explain the Global Real Interest Rate

As the above analysis finds that the global real interest rate has moved over time and currently *does* appear to be high by historical standards, the question arises of why this might be so. Empirical research on the determinants of the global real interest rate has split into structural and reduced-form modeling approaches. Barro and Sala-i-Martin (1990) and Barro (1992) follow the first route and model the short-term global real interest rate as the price that equilibrates worldwide investment and saving.^{11} Their specification of the investment and saving ratios is such that both ratios depend on a large, unexplained, persistent part and a short-term, business cycle part driven by stock prices, the relative price of oil, and changes in money supply growth and in fiscal policy. They conclude that stock markets and oil prices play a major role in explaining real interest rates, and that world monetary growth and public debt exert only a secondary influence.^{12} Barro’s (1992) results indicate that a 1 percentage point increase in the ratio of world public debt to GDP raises the world real short-term interest rate by 12 basis points.

Most other empirical studies use reduced-form equations of the form

where *d _{t}* and

*g*denote government debt and government consumption, both normalized by GDP. The studies differ as to whether national real interest rates or global real interest rates were taken as the dependent variables (Evans (1987), Ford and Laxton (1995), and Howe and Pigott (1991)), whether other arguments were included in the reduced form (see, for example, Howe and Pigott (1991) and Coorey (1991)), and whether ε

_{t}*, is assumed to be an independently and identically distributed residual (Ford and Laxton (1995)) or a stationary variable with zero mean only (Coorey (1991), Evans (1987), and Howe and Pigott (1991)).*

_{t}Ford and Laxton (1995) estimate a system of equations of the form

where *d _{t}* and

*g*denote the normalized net debt and normalized government purchases for all Organization for Economic Cooperation and Development (OECD) countries. Their specification is similar to that of Gagnon and Unferth (1993), except that the global real interest rate is replaced by that part of the rate explained by world fiscal aggregates. They find that a 1 percentage point increase of the ratio of net public debt to GDP in the OECD area raises interest rates by about 30 basis points. Ford and Laxton’s work is unique in that they assume

_{t}*ε*to be a white-noise process.

_{it}In most of the other empirical studies, equations analogous to equation (7) represent a long-run relationship only. Evans (1987), for example, estimates cointegration equations for nominal three-month U.S. treasury bill rates using both the inflation rate and U.S. fiscal variables as regressors; he does not find any stationary residual for this kind of equation. Furthermore, his coefficients on *d _{t}* are always negative and quite large for the period from 1948 to 1986.

^{13}Coorey (1991) estimated long-run equations for various U.S. real interest rates using a larger set of variables, including fiscal and monetary policy, demographic, wealth, and productivity variables. She also obtained sizable fiscal policy effects, although the strength of the effect depends upon the exact specification.

Howe and Pigott (1991) use a two-step process to estimate the determinants of real interest rates for five industrial countries. They use cointegration equations to define what they call the “equilibrium real interest rate” for each country as a function of the rate of return on business capital, total nonfiancial debt relative to GDP, and the relative riskiness of bonds. They then use an error-correction model to estimate the actual changes in real interest rates as a function of the deviations between the calculated real interest rates and the equilibrium rates, as well as of fiscal and monetary policy factors and business cycle factors. They find small but significant effects of government debt on real interest rates: a 1 percentage point increase in the total debt (government debt and total nonfinancial debt) raises real interest rates between 3 basis points (in the United States) and 30 basis points (in France). A limitation of their work is that they do not take into account linkages among countries.

Brunner and Kaminsky (1994) estimate a structural vector autoregression (VAR) model for their (derived) global real interest rate using fiscal deficits as a fraction of GDP and the world money supply growth in Germany, Japan, and the United States as explanatory variables. They find that U.S. fiscal shocks and an unexplained residual, which they call a “worldwide real shock,” account for most of the variation in global real interest rates over the past 15 years.

### A Modeling Framework for the Global Real Interest Rate

The global real interest rate is best seen as determined through a framework of demand and supply for assets. If all countries that are part of the integrated world goods and capital markets are viewed as a single entity, the global real interest rate should be determined by world aggregates. One of the critical conceptual problems in isolating the main determinants of real interest rates is the simultaneity among asset markets, whereby equilibrium is determined for all assets at the same time. However, it is possible to isolate dominant common factors in all asset demand functions, such as wealth, for example, and to focus on their impact on real interest rates.

The starting point for the modeling is a framework similar to that used in Appendix I. There are households that maximize their utility from consumption over two periods. Unlike in the model of Appendix I, however, it is now assumed that there are overlapping generations—that is, households exist for two periods and work only in the first period. Each household receives some wage income, which, after taxes, is either consumed or invested in risky stocks and risk-free government bonds. The gross return on this investment is the source for this second-period consumption. Firms invest so as to maximize the value of firms or, in other words, to maximize the value of the outstanding shares that represent claims on the return on the capital stock. Uncertainty is resolved in the beginning of each period, so that new households know about their labor income and the marginal productivity of capital. Future marginal products of capital are uncertain because of productivity shocks.

In this framework, new households consume a fraction of their net wealth in the first period. Their asset demands are also a function of net wealth, which is equal to the value of the initial labor income (net of taxes) and other initial wealth. The level of their asset demand, as well as the fraction of their assets held in risk-free bonds, depends on the real interest rate, as well as on the real return on the world stock portfolio (*r ^{s}*). Therefore, the bond demand function of new households in country

*i*can be written as

where the signs on top of a variable indicate the sign of the partial derivative with respect to that variable, and where *a ^{i}* denotes the net wealth of an investor in country

*i*in terms of world output. (All variables are in a common currency.) An increase in national real interest rates raises asset demand in general if it is assumed that the substitution effect dominates the income effects (including discounting of future income), so that present consumption becomes expensive relative to future consumption. A real interest rate increase also raises the demand for bonds relative to that for stocks.

^{14}An increase in world stock returns relative to the real interest rate decreases the demand for bonds. An increase in net wealth increases asset demand because investors will consume only part of that increase in the current period.

For the discussion of the effects of fiscal policy, it is important to know whether so-called Ricardian equivalence holds. If the equivalence holds, investors do not include outstanding government debt in their net wealth as they anticipate future tax increases. If Ricardian equivalence does not hold, a fraction of the outstanding debt, ø, is part of investors’ net wealth.^{15} It is most likely that Ricardian equivalence holds to some degree, that is -1 < ø < 1, because investors anticipate that they will suffer from some of the future tax increases.^{16} Net wealth can therefore be decomposed into a private component, h, and the government component, *øb ^{i}* where

*b*denotes government debt as a fraction of GDP in country i. The bond demand function of an investor in country

^{i}*i*can therefore be rewritten as

In equilibrium, only government bonds can be in positive net supply. The supply of government bonds, for simplicity, is assumed to be strictly exogenous.^{17} If it is also assumed that all government bonds issued by each national government, denoted by

The assumptions that technology shocks are the sole determinant of the return on capital (given the level of technology) and that there are integrated capital markets allow the model to be solved recursively. Given the world real interest rate, stock returns are determined by net investment, which, in turn, is a function of the level of technology and the expected technology shock, summarized in the variable 9. It can be shown that stock returns also depend positively on real interest rates. Therefore, the equilibrium condition (10) can be written as^{18}

Equilibrium is represented in Chart 4, The *bb* schedule represents the exogenous supply of government bonds, whereas the upward sloping locus *dd* portrays demand. The slope of this locus depends on the Ricardian parameter ø. If it is equal to one, that is, if households fully take into account the future tax implications of the current government debt, the locus is horizontal for all values of *b*, so that an increase in government debt has no effect on interest rates. If the parameter øis smaller than one, an increase in government debt leads to a rise in the real interest rate, as the shift in the *bb* schedule to *b’b’* demonstrates.

**The Global Real Interest Rate (r) and an Increase in Government Debt (b)**

**The Global Real Interest Rate (r) and an Increase in Government Debt (b)**

**The Global Real Interest Rate (r) and an Increase in Government Debt (b)**

This model is an oversimplification in two dimensions. First, by assuming the existence of a world stock market, and by assuming that various countries’ government bonds are perfect substitutes, a degree of capital mobility is assumed that is not yet realized in today’s world economy. Second, the model focuses only on long-run equilibria, abstracting from short-term influences, such as monetary policy and the business cycle. It is clear that, in the short term, real interest rates are determined by other factors besides the level of outstanding government debt (and private net wealth) and stock returns. Monetary policy instruments, for example, have an impact on real interest rates in the short run because of various frictions that prevent prices from adjusting immediately or within a short time span. But it is also well established that the effects of monetary tightening or easing are only transitory, and that, in the long run, monetary policy mainly determines only the price level (see Friedman (1968)). Real movements of interest rates are affected only through risk premiums (the covariation of inflation with consumption). These premiums are most likely to be small in the countries included in this sample, and their sign is by no means certain (see Labadie (1989)).

### Model Specification

Because asset demands are interrelated through considerations of risk and return, a proper structural model would require the estimation of a whole system of asset demand and supply functions, as well as the evolution of aggregate wealth. As such an estimation is beyond the scope of this study, a reduced-form modeling strategy has been chosen instead. The general two-step modeling procedure described by Howe and Pigott (1991) has been followed; however, this procedure has been applied to the concept of a global real interest rate rather than a set of national rates. This choice is based on the results of the unit root tests in Table 1, which show no mean reversion in global real interest rates in the sample period. This study does not assume that the global real interest rate literally follows a random walk process that is without bounds, but rather that there are mean shifts—owing to preference shifts, shifts in demographics, and so forth—that occur too frequently to be handled with dummy variables.

The first step is to estimate the equilibrium global real interest rate as a function of the variables described above. Because the equilibrium global real interest rate is a long-run concept, it can be thought of as the natural real interest rate that would occur if there were no transitory determinants of the real interest rate, such as monetary policy and the business cycle. It is important that the equilibrium function generate stationary residuals. The second step is to account for transitory movements in the global real interest rate owing to factors such as monetary policy shifts or business cycle factors.

In the approach followed in this study, the equilibrium global real interest rate is determined by the return on capital in the real economy and the outstanding public debt. It is assumed that other factors are captured by a constant and a stationary error process. This approach raises the question of how to measure the return on capital in the real sector, and of which public debt variable to use. Two natural candidates for the variable for return on capital in the real sector are the return on the capital stock derived from national accounts statistics and the real aggregate stock market returns. Both variables measure average returns. From a theoretical perspective, the marginal rate of returns would be preferable; however, from a long-run perspective, this should be only a minor problem because the two measures should coincide. Stock market returns are appealing because they represent a financial market return and reflect the market valuation of physical and human capital.

From an asset demand and supply perspective, gross public debt seems to be the relevant variable because it represents the outstanding stock of bonds. One might argue, however, that only the net outstanding public debt matters for the non-Ricardian effects of debt on interest rates. The answer to the question of whether gross or net debt should be used depends on the definition of these aggregates. It is clearly preferable to use a debt aggregate that includes only the consolidated liabilities of the general government sector (that is, liabilities of both central and local governments). It is less certain whether one should use debt measures that include the consolidated liabilities of the general government and the social security sectors. As, most likely, future taxes will also have to cover liabilities of the social security sector, a gross measure that does *not* include consolidated liabilities of the general government and the social security sector seems preferable. For this reason, gross public debt figures compiled by the OECD have been used.

Although stock returns and gross public debt are preferable as long-run determinants of real interest rates, equation (12) uses other arguments as explanatory variables to facilitate comparison with other studies, as follows:

where *B* denotes government debt (in current prices), *NGDP*, the GDP in current prices, and

These equations are only semireduced forms. The return on capital depends partly, as shown above, on the real interest rate because of the relationship between risk and return in asset markets. This might cause some simultaneity problems. If equation (12) is thought of as a cointegration relationship, the simultaneity question might be less acute. The asymptotic bias is known to be zero, although the finite sample bias can be considerable. This is taken into account by using Stock and Watson’s (1993) dynamic ordinary least squares estimator. Also, an advantage of the cointegration approach is that actual debt figures can be used because targeted and actual debt should coincide in the long run.

The second step of the modeling process, the modeling of the transitory movements, must deal with both a short- and a long-term equilibrium global interest rate. From a theoretical perspective, the long-term equilibrium value for both should be equal because of term structure linkages.^{19} Therefore, the long-term equilibrium values for long-term global interest rates are used to explain both the short- and the long-term deviations from equilibrium. The transitory dynamics are modeled using the following error-correction models:

where variables in the vector *X _{t}* capture business cycle factors as well as monetary and fiscal policy variables that have a transitory impact on real interest rates, and where

### Empirical Results

Only annual data were available for the return on capital and the outstanding public debt variables, so all the regression results are based on annual data. The stationarity of the residuals was tested using both augmented Dickey-Fuller and Philips-Perron tests. The coefficients were estimated using Stock and Watson’s lead-lag estimator to correct for finite sample biases owing to the simultaneity problem. The asymptotic standard errors were calculated using an autocorrelation-consistent variance-covariance matrix for the coefficients. Given the relatively short sample (1960-93), the number of leads and lags was restricted to one on a priori grounds.

The results for both short- and long-term global real interest rates are presented in Table 3.^{20} The short-term global real interest rate based on GDP weights was used as the dependent variable. The Philips-Perron unit root tests allow for the rejection of the null hypothesis of no cointegration. The coefficients for the public debt variables have the right sign and are always significant. The long-term effects on global real interest rates of an increase in the ratio of gross public debt to GDP of 1 percentage point vary between 16 and 20 basis points. For each percentage point rise in the ratio, net public debt generates larger interest rate effects—between 37 basis points and 55 basis points—than gross public debt. With the ratio of gross public debt to GDP as the fiscal variable, the coefficients on the rate of return on capital, as well as on stock market returns (the index of capital gains on stock), are both significant and of the right sign. The coefficient on the return on capital has the wrong sign if the ratio of net public debt to GDP is used as a regressor, whereas the coefficient on stock market returns has the right sign but is insignificant. Only the cointegration estimations using gross public debt as a ratio to GDP seem to be robust, with a significant coefficient with the right sign.

**Cointegration Regressions for Global Real Interest Rates, 1960-93 ^{1}**

*(Coefficients and asymptotic errors based on Stock and Watson—s (1993) dynamic ordinary least squares estimator)*

^{1}Annual data.

^{2}Weighted average.

^{3}Capital income in the business sector as percentage of capital slock.

^{4}Percentage changes in stock prices.

^{5}Augmented Dickey-Fuller unit root test.

^{6}Philips-Perron unit root test. Three asterisks denote rejection of the null hypothesis of a unit root in the residuals at the 1 percent level, two asterisks at the 5 percent level, and one asterisk at the 10 percent level.

**Cointegration Regressions for Global Real Interest Rates, 1960-93 ^{1}**

Short-Term Global Real Interest Rate^{2} | Long-Term Global Real Interest Rate^{2} | ||||||||
---|---|---|---|---|---|---|---|---|---|

Return on capital^{3} | 0.663 | -0.427 | 0.534 | -0.396 | |||||

(0.196) | (0.208) | (0.197) | (0.233) | ||||||

Capital gains on stock index^{4} | 0.162 | 0.071 | 0.210 | 0.196 | |||||

(0.047) | (0.077) | (0.034) | (0.053) | ||||||

Gross government debt | 0.195 | 0.1574 | 0.129 | 0.102 | |||||

(in percent of GDP) | (0.038) | (0.032) | (0.065) | (0.029) | |||||

Net government debt | 0.550 | 0.372 | 0.421 | 0.161 | |||||

(in percent of GDP) | (0.083) | (0.117) | (0.090) | (0.089) | |||||

ADF t-statistics^{5} | -2.437 | -2.759 | -2.967 | -2.284 | -1.798 | -2.522 | -3.588 | 2.968 | |

PP t-statistics^{6} | -3.301** | -3.778* | -3.528* | -3.889** | -4.724*** | -3.920** | -3.467 | 3.792* |

^{1}Annual data.

^{2}Weighted average.

^{3}Capital income in the business sector as percentage of capital slock.

^{4}Percentage changes in stock prices.

^{5}Augmented Dickey-Fuller unit root test.

^{6}Philips-Perron unit root test. Three asterisks denote rejection of the null hypothesis of a unit root in the residuals at the 1 percent level, two asterisks at the 5 percent level, and one asterisk at the 10 percent level.

**Cointegration Regressions for Global Real Interest Rates, 1960-93 ^{1}**

Short-Term Global Real Interest Rate^{2} | Long-Term Global Real Interest Rate^{2} | ||||||||
---|---|---|---|---|---|---|---|---|---|

Return on capital^{3} | 0.663 | -0.427 | 0.534 | -0.396 | |||||

(0.196) | (0.208) | (0.197) | (0.233) | ||||||

Capital gains on stock index^{4} | 0.162 | 0.071 | 0.210 | 0.196 | |||||

(0.047) | (0.077) | (0.034) | (0.053) | ||||||

Gross government debt | 0.195 | 0.1574 | 0.129 | 0.102 | |||||

(in percent of GDP) | (0.038) | (0.032) | (0.065) | (0.029) | |||||

Net government debt | 0.550 | 0.372 | 0.421 | 0.161 | |||||

(in percent of GDP) | (0.083) | (0.117) | (0.090) | (0.089) | |||||

ADF t-statistics^{5} | -2.437 | -2.759 | -2.967 | -2.284 | -1.798 | -2.522 | -3.588 | 2.968 | |

PP t-statistics^{6} | -3.301** | -3.778* | -3.528* | -3.889** | -4.724*** | -3.920** | -3.467 | 3.792* |

^{1}Annual data.

^{2}Weighted average.

^{3}Capital income in the business sector as percentage of capital slock.

^{4}Percentage changes in stock prices.

^{5}Augmented Dickey-Fuller unit root test.

^{6}Philips-Perron unit root test. Three asterisks denote rejection of the null hypothesis of a unit root in the residuals at the 1 percent level, two asterisks at the 5 percent level, and one asterisk at the 10 percent level.

The results for the long-term real interest rate are very similar to those for the short-term rate. The coefficients on the rate of return on business capital are of similar magnitude and significant as well. As with the short-term rate, these coefficients also have the wrong sign if the ratio of net public debt to GDP is used as a regressor. The coefficients on the stock market returns have the right sign, are roughly similar, and are significant independent of the debt measure used. These coefficients are also larger in size than with the short-term rate, indicating that the link between stock and bond markets is more important in the market for bonds with longer maturities. The results also indicate that the effects of government debt on real interest rates are smaller if long-term yields are used. These estimates on the long-term effects of government debt are close to those of Barro (1992) in the case of gross government debt, and close to those of Ford and Laxton (1995) in the case of net public debt (with stock market returns used as regressors).

Some versions of equation (12) have also been estimated with national real interest rates as left-hand variables to show that the global real interest rate is a significant factor in their determination. The long-run level of national real interest rates is explained by a world return variable, a world government debt measure, and a country-specific constant. The results confirm that world aggregates can explain national real interest rates reasonably well, based on the Philips-Perron unit root test on the residuals. All equations display cointegration at the 10 percent confidence level. The coefficients have the right sign, with the return on capital and gross government debt as explanatory variables for all countries except the Netherlands. The rate of return on stocks is more problematic. The effects expected for long-term interest rates hold in seven out of the eight countries; however, they hold for short-term real interest rates in only four out of the eight countries.

Cointegration results have to be interpreted with some caution in small samples.^{21} Given the considerable degree of uncertainty concerning the size of the long-run coefficients, the study also used Johansen’s maximum likelihood estimator to test for cointegration and to estimate the long-run coefficients. All results strongly support the existence of a single unique cointegration vector. The most noteworthy aspect of the estimated cointegration vectors is how close the coefficients for stock returns and gross government debt (as a fraction of GDP) are both for short-term and long-term global real interest rates. The estimates for the long-run coefficients on stock return lie in the interval [.2, .3] and the estimates for the coefficient on gross government debt in the interval [.08, .11]. Also the coefficients on debt depend less on which debt measure is used, as both gross and net world public debt produce similar long-run effects. A final consideration is the possibility that each variable is weakly exogenous when it comes to the estimation of the cointegration vectors. Gross public debt is weakly exogenous in almost all cases. The results are much more ambiguous when it comes to real interest rates and real stock returns. The real stock return is almost never weakly exogenous, whereas there is some indication that global real interest rates are. However, these results are quite sensitive to specification issues, such as the lag length and the inclusion of a constant in the cointegration vector, and strong conclusions cannot be drawn.

These results show that much of the long-term variation in real interest rates can be explained by rates of return in the real economy and by the ratio of gross government debt to GDP. The link between rates of return and long-term variation in real interest rates is less solid, although the relation is stronger for long-term bond yields. The results also suggest that the estimates for the long-term real interest rates come closer to the underlying equilibrium relationship than do those for the short-term real interest rates, as the coefficients in the former regressions are more in line with theory.

The final regression results concern the transitory deviations of the short- and long-term global real interest rates from the long-term equilibrium real interest rate. Table 4 reports the regression results for the weighted-average global real interest rates.^{22} Only the reduced equations, that is, equations that underwent the reduction from the general to the specific, are shown. (The first and second lags of all variables, as well as the current values for the change in the real discount rate and the GDP growth rate, are included in the general specifications.) For the short-term rate, monetary and fiscal policies, measured by changes in the world real discount rate and the ratio of world budget deficits, are significant determinants, as well as the error-correction adjustment, the lagged long-term changes, and the changes in the return on stocks. The world GDP growth rate and changes in the long-term global real interest rate were insignificant.^{23} The results for the long-term global real interest rate were remarkably different. Monetary and fiscal policies have no significant direct impact. But the business cycle, as measured by the growth in world GDP, is a primary determinant of the short-term dynamics. Fiscal and monetary policies still influence the long-term global real interest rate, however, through the lagged change in short-term real interest rates and the error-correction term.

**Error-Correction Models for Global Real Interest Rates**

(Standard errors in parentheses)

^{1}Weighted average.

^{2}Lagrange multiplier lest of residual first order serial correlation.

**Error-Correction Models for Global Real Interest Rates**

(Standard errors in parentheses)

Changes in Global Real Interest Rate^{1} | ||
---|---|---|

Short term | Long term | |

Independent variables | ||

Lagged changes in short-term real interest rate | 0.377 | |

(0.077) | ||

Lagged changes in long-term global real interest rate | 0.210 | 0.275 |

(0.071) | (0.084) | |

Current changes in real world discount rate | 0.945 | |

(0.049) | ||

Lagged changes in debt-to-GDP ratio | -5.124 | |

(2.515) | ||

Current growth rate of world GDP | 0.246 | |

(0.037) | ||

Lagged growth rate of world GDP | -0.270 | |

(0.037) | ||

Lagged change in stock returns | 0.001 | |

(0.003) | ||

Lagged deviation from equilibrium value | -0.153 | -0.119 |

(0.041) | (0.032) | |

Fit statistics | ||

0.937 | 0.775 | |

0.382 | 0.477 | |

AR (1)^{2} | 0.25 | 1.64 |

^{1}Weighted average.

^{2}Lagrange multiplier lest of residual first order serial correlation.

**Error-Correction Models for Global Real Interest Rates**

(Standard errors in parentheses)

Changes in Global Real Interest Rate^{1} | ||
---|---|---|

Short term | Long term | |

Independent variables | ||

Lagged changes in short-term real interest rate | 0.377 | |

(0.077) | ||

Lagged changes in long-term global real interest rate | 0.210 | 0.275 |

(0.071) | (0.084) | |

Current changes in real world discount rate | 0.945 | |

(0.049) | ||

Lagged changes in debt-to-GDP ratio | -5.124 | |

(2.515) | ||

Current growth rate of world GDP | 0.246 | |

(0.037) | ||

Lagged growth rate of world GDP | -0.270 | |

(0.037) | ||

Lagged change in stock returns | 0.001 | |

(0.003) | ||

Lagged deviation from equilibrium value | -0.153 | -0.119 |

(0.041) | (0.032) | |

Fit statistics | ||

0.937 | 0.775 | |

0.382 | 0.477 | |

AR (1)^{2} | 0.25 | 1.64 |

^{1}Weighted average.

^{2}Lagrange multiplier lest of residual first order serial correlation.

## Conclusions

Until recently, it was probably sufficient to look at trends in national interest rates to get a sense of the pressures that ex ante investment demands place on the world supply of saving. With the increasing integration of world capital markets, however, it now makes more sense to think about the concept of a global real interest rate that reflects global financial forces. Although several methodologies can be employed to measure such a global real interest rate—a weighted-average approach, a principal components approach, and a common trend approach—the analysis in this paper suggests that they all indicate the same basic trend over the past third of a century. Global real interest rates averaged 1-2 percent in the 1960s and early 1970s, roughly zero in the oil shock period of 1973-1980, and a much higher 3-4½ percent in the 1981-94 period. This paper’s model of shifts in global interest rate regimes finds strong empirical support for such a breakdown.

These findings raise the obvious question of why real interest rates have been so much higher over the past decade and a half than they were from 1960 to 1972, which can be considered to be a useful reference period. The paper’s modeling structure suggests that increased levels of world government debt relative to GDP since the early 1980s are the most important factor in the increase in the global real interest rate. This increase in debt explains a larger share of the increase in real rates than other commonly posited factors, including the general worldwide increase in the rate of return on productive capacity. Because the real interest rate is such an important determinant of economic growth, and because higher interest rates are associated with lower levels of capital formation, productivity, and income growth, governments worldwide need to be concerned about the repercussions of increases in not only their own ratios of debt to GDP, but also their neighbors’.

## Appendix I Common Components in Global Real Interest Rates

In this appendix, a two-period, two-country asset-pricing model is used to provide a theoretical basis for the discussion of global real interest rates in the first section of the paper. Assuming that each of the two countries is populated with identical consumers whose preferences might be country specific, consumers in both countries face two decisions. The first concerns the intertemporal allocation of spending over the two periods; the second concerns the temporal allocation between traded and nontraded goods. The real spending of a consumer in country *i (i* - 1,2) in period *t* (*t* = 0,1) is denoted as

where β* ^{i}* denotes the inverse of one plus the rate-of-time preference. The consumer’s spending is allocated between traded goods

with spending restricted by the requirement that the present value of consumption expenditures equal the present value of the consumer’s endowment over the two periods

where *i* receives in period *t*, *t*, and the consumption price index in period *t*. *i* buys in period 0. One bond can be acquired with one unit of the tradable good and yields a net return of *r* units of the tradable good in period 1.

In equilibrium, the consumption of nontraded goods must be equal to the endowment of nontradables in each period and in each country. That is,

Also, the total consumption of tradables must be equal to the total endowment of tradables in each period, that is,

Finally, the net supply of bonds has to be equal to zero:

The endowments * _{ij}* and variance σ

*ij*, where

*j*=

*X, Y*. Given the notation above, it also follows that consumers in both countries are assumed to know the state of the world in period 0 when they form their decisions or, in other words, to have learned about the realizations of the endowment shocks in that period.

The intertemporal spending decision is governed by the following familiar first-order condition:

which requires the marginal rate of substitution of spending to be equal to the marginal rate of transformation, that is, the return on savings. Using the goods market equilibrium conditions in this first-order condition leads to the equation^{24}

which defines the demand for bonds for consumers who are price takers in world capital markets (*X ^{i}*

_{0}). The bond demand functions

The bond market equilibrium condition (equation (A6)), finally, leads to the following function for the equilibrium global real interest rate:

The determination of real interest rates in this setup is standard and will not be discussed in more detail; instead, the focus will be on only the aspects relevant for the measurement of the global real interest rate,

There is a well-defined world real interest rate in this setup because there is an asset denominated in tradable goods that allows consumers to smooth their consumption of tradable goods. In empirical work, only so-called consumption-based real interest rates, that is, nominal interest rates deflated by some consumer price index, are available for most countries. This model, however, can be used to analyze the determination of the consumption-based real interest rate and to discuss the implications for empirical work. The gross consumption-based real interest rate for country *i* is defined (according to Frenkel and Razin (1992, p. 140)) as

Taking natural logarithms, the net real interest rate can be approximated by

where *q ^{i}* denotes the percentage change in the consumption price index

This equation clearly illustrates the role of the consumption-based real interest rate: it determines the intertemporal allocation of spending. Households take the expected price change in the consumption price index into account when they decide on their intertemporal allocation.

It can be shown that the consumption price index

and that in equilibrium the relative price of nontraded goods is

Using the fact that the consumption of traded goods is a function of the same factors as the bond demand function (equation (A9)) because of the trade balance constraints, that is,

allows one to write the change in the consumption price index *q* as

Consequently, the consumption-based real interest rate of country *i* is a function such that

Equation (A18) is the foundation of the claim that there are common factors in national consumption-based real interest rates that can be extracted to construct a global real interest rate. The country-specific shocks, that is, shocks to the endowments of nontradable goods *r*, but they do have an impact on the consumption-based real interest rates ^{25} It is clear from equations (A17) and (A18) that both the interest rate *r* and the expected real exchange rate changes *q* move with common components, that is

What are the consequences of using consumption-based real interest rates in empirical work? The common component picks up the impact of shocks that are transmitted to both countries, in particular, the impact of the realized shocks *B*, but also expected changes in real exchange rates. If there were no shocks to the endowments of nontraded goods, this fact would not constitute any problem because the common movements in the expected ratios of tradable consumption in period 0 and period 1, that is, *C _{0}/C_{1}*, are exactly offset across the two countries (note that these ratios are perfectly negatively correlated in this model). Shocks to the endowments of nontradables also induce real exchange rate changes, however; these effects are not necessarily related to other shocks nor can they be expected to be offset. It follows, therefore, that the global real interest rate, defined as the common component in consumption-based real interest rates, should be seen as an average real interest rate on world spending growth rather than on the growth of spending in tradable goods alone.

^{26}However, this average interest rate is not a real price in the sense of a decision variable because no underlying asset exists.

^{27}

It is argued in the text that a global real interest rate is an interest rate on world spending or, in equilibrium, on world output. This claim can be illustrated by rewriting the first-order conditions in terms of world spending:

where Ω denotes the growth rate of world GDP, that is the sum of all traded and nontraded goods, *S _{t}* denotes the consumption price index of country

*i*in terms of the country in which world output/spending is measured, that is, the ratio

## Appendix II Modeling Inflation Expectations

Measuring real average bond yields to maturity requires an inflation forecast for the same period as the bond. Blanchard and Summers (1984) propose a duration-weighted average of expected future inflation rates. Given that average yields to maturity are used for long-term government bond rates, the question arises of which forecast horizon to choose. The answer depends on the time-series properties of inflation. If inflation is nonstationary and can be modeled as a random walk, the problem of the forecast horizon is irrelevant. If inflation can be approximated by an autoregressive moving average (ARMAf (p, q)) model, choosing the forecast horizon depends on the parameters of the model.

The three-month inflation rates in all countries included in the sample display unit root or near unit root properties over the sample period.^{28} Any recursive univariate model, therefore, will lead to forecasts in which innovations in the current inflation rate are incorporated as (almost) permanent shifts.^{29} This feature turns out to have rather problematic implications for the real long-term interest rates in the years 1973-74 and 1979. It seems rather unlikely, however, that market participants expected oil-shock-induced innovations in the inflation rates to be permanent. Surveys of inflation expectations show that, although there is a certain degree of inertia in these expectations, they are updated with respect to events in the recent past. It is also likely that agents learn and react differently in different monetary policy regimes. For that reason, nonlinear univariate time-series models might capture the process of inflation expectation formation more accurately than linear time-series models. However, an adaptive expectations formula comes close to observed expectation formation patterns. Given the near-random-walk properties of inflation, this procedure also generates optimal linear forecasts (see Muth (1961)). Adaptive expectations were generated by using the exponential-smoothing formula

where π_{t}, denotes the annual inflation rate and

## Appendix III Constructing Global Real Interest Rates

In this appendix, the construction of a global real interest rate using the principal component and the common trend estimation approaches are documented.

### Principal Components Approach

The first principal components of standardized short- and long-term real interest rates were calculated for the period from 1960:QI to 1994:QIII. Global real interest rates were then generated as a weighted average of national real interest rates, with weights based on the load factors of the first principal component of standardized national real interest rates.^{30} Global real interest rates based on both eight and three countries were calculated. The results of the principal component analysis can be found in Tables A2 and A3. The first principal component explains roughly half of the variation in short-term real interest rates and about 65 percent in long-term real interest rates. The additional explanation by other principal components declines rapidly, although the importance of country-specific factors is remarkable.^{31} It is interesting that the weights implied by the load factors are much more equal than the GDP weights.

**Summary Statistics for National Real Interest Rates**

^{1}In percent; standard deviation in parentheses.

^{2}Augmented Die key-Fuller unit root test. Automatic lag length selection according to significance level of last lag of lagged first difference of series. Initial lag length is six lags (lag length in parentheses).

^{3}Minimum t-statistics from a series of augmented Dickey-Fuller regressions with a 1-0 dummy variable in every *period* (excluding observations near end points). Lag lengths in parentheses. Critical values of test statistics taken from Perron and Vogelsang (1992).

^{4}Period in which minimum *t*-statistics occurs.

^{5}Three asterisks denote rejection of the null hypothesis of a unit root in the residuals at the 1 percent level, two asterisks at the 5 percent level, and one asterisk at the 10 percent level.

**Summary Statistics for National Real Interest Rates**

Mean^{1} | Break | ||||||||
---|---|---|---|---|---|---|---|---|---|

60:QI-94:QIII | 60:QI-72:QIV | 73:QI-80:QIV | 8l:QI-94:QIII | ADF t-statistics I^{2, 5} | ADF t-statistics II^{3, 5} | Date^{4} | |||

Short-term interest rates | |||||||||

Belgium | 3.019 | 1.322 | 0.950 | 5.829 | -1.627 (6) | -3.472 (6) | 76:QI | ||

(3.751) | (2.612) | (4.643) | (1.943) | ||||||

Canada | 3.073 | 2.235 | -0.059 | 5.686 | -2.278 (3) | -7.047 (0) | *** | 80:QIII | |

(3.281) | (2.243) | (8.173) | (4.622) | ||||||

Germany | 2.893 | 1.995 | 1.988 | 4.268 | -3.715 (4) | *** | -5.657 (5) | *** | 79:Q1I |

(2.688) | (2.698) | (2.946) | (1.850) | ||||||

Japan | 1.396 | 0.556 | -1.365 | 3.796 | -3.476 (4) | ** | -5.986 (3) | *** | 73:QIV |

(5.037) | (3.231) | (7.352) | (2.688) | ||||||

Netherlands | 1.814 | -0.311 | 0.469 | 4.807 | -2.606 (4) | -4.981 (4) | ** | 78:QII | |

(4.186) | (5.727) | (4.001) | (2.126) | ||||||

Switzerland | 0.084 | -0.807 | -1.520 | 1.860 | -2.326 (3) | -4.313 (3) | 73:QI | ||

(3.166) | (2.586) | (3.777) | (2.367) | ||||||

United Kingdom | 1.050 | 0.746 | -1.832 | 4.760 | -2.420 (3) | -4.206 (4) | 80:QI | ||

(6.188) | (3.290) | (7.915) | (4.132) | ||||||

United States | 1.281 | 1.202 | -1.942 | 3.225 | -2.093 (2) | -3.761 (1) | 79:QIV | ||

(2.849) | (1.113) | (4.585) | (6.421) | ||||||

Long-term interest rates | |||||||||

Belgium | 3.834 | 3.521 | 1.176 | 5.676 | -1.778 (4) | -3.261 (4) | 77:QIII | ||

(2.492) | (0.896) | (3.373) | (0.922) | ||||||

Canada | 3.559 | 3.411 | 0.701 | 5.362 | -1.379 (5) | -3.765 (1) | 82:QIV | ||

(2.135) | (0.574) | (1.373) | (1.442) | ||||||

Germany | 4.026 | 44.74 | 2.932 | 4.524 | -2.177 (1) | -3.131 (3) | 82:QIV | ||

(1.154) | (0.841) | (0.842) | (1.183) | ||||||

Japan | 2.490 | 3.720 | -1.937 | 3.903 | -2.032 (4) | -3.474 (3) | 77:QII | ||

(3.812) | (3.034) | (4.628) | (1.002) | ||||||

Netherlands | 2.947 | 1.597 | 1.399 | 5.124 | -1.553 (4) | -3.891 (3) | 77:QII | ||

(2.285) | (1.079) | (2.163) | (1.248) | ||||||

Switzerland | 0.889 | 0.779 | 0.149 | 1.418 | -2.866 (6) | * | -3.964 (6) | 82:QIV | |

(2.658) | (1.242) | (1.768) | (1.184) | ||||||

United Kingdom | 2.303 | 2.746 | -1.732 | 4.233 | -2.178 (6) | -3.577 (4) | 77:QI | ||

(3.265) | (1.489) | (3.916) | (1.650) | ||||||

United States | 3.410 | 2.863 | 0.728 | 5.487 | -2.307 (3) | -5.372 (3) | *** | 80:QI1 | |

(2.237) | (0.386) | (1.327) | (1.640) |

^{1}In percent; standard deviation in parentheses.

^{2}Augmented Die key-Fuller unit root test. Automatic lag length selection according to significance level of last lag of lagged first difference of series. Initial lag length is six lags (lag length in parentheses).

^{3}Minimum t-statistics from a series of augmented Dickey-Fuller regressions with a 1-0 dummy variable in every *period* (excluding observations near end points). Lag lengths in parentheses. Critical values of test statistics taken from Perron and Vogelsang (1992).

^{4}Period in which minimum *t*-statistics occurs.

^{5}Three asterisks denote rejection of the null hypothesis of a unit root in the residuals at the 1 percent level, two asterisks at the 5 percent level, and one asterisk at the 10 percent level.

**Summary Statistics for National Real Interest Rates**

Mean^{1} | Break | ||||||||
---|---|---|---|---|---|---|---|---|---|

60:QI-94:QIII | 60:QI-72:QIV | 73:QI-80:QIV | 8l:QI-94:QIII | ADF t-statistics I^{2, 5} | ADF t-statistics II^{3, 5} | Date^{4} | |||

Short-term interest rates | |||||||||

Belgium | 3.019 | 1.322 | 0.950 | 5.829 | -1.627 (6) | -3.472 (6) | 76:QI | ||

(3.751) | (2.612) | (4.643) | (1.943) | ||||||

Canada | 3.073 | 2.235 | -0.059 | 5.686 | -2.278 (3) | -7.047 (0) | *** | 80:QIII | |

(3.281) | (2.243) | (8.173) | (4.622) | ||||||

Germany | 2.893 | 1.995 | 1.988 | 4.268 | -3.715 (4) | *** | -5.657 (5) | *** | 79:Q1I |

(2.688) | (2.698) | (2.946) | (1.850) | ||||||

Japan | 1.396 | 0.556 | -1.365 | 3.796 | -3.476 (4) | ** | -5.986 (3) | *** | 73:QIV |

(5.037) | (3.231) | (7.352) | (2.688) | ||||||

Netherlands | 1.814 | -0.311 | 0.469 | 4.807 | -2.606 (4) | -4.981 (4) | ** | 78:QII | |

(4.186) | (5.727) | (4.001) | (2.126) | ||||||

Switzerland | 0.084 | -0.807 | -1.520 | 1.860 | -2.326 (3) | -4.313 (3) | 73:QI | ||

(3.166) | (2.586) | (3.777) | (2.367) | ||||||

United Kingdom | 1.050 | 0.746 | -1.832 | 4.760 | -2.420 (3) | -4.206 (4) | 80:QI | ||

(6.188) | (3.290) | (7.915) | (4.132) | ||||||

United States | 1.281 | 1.202 | -1.942 | 3.225 | -2.093 (2) | -3.761 (1) | 79:QIV | ||

(2.849) | (1.113) | (4.585) | (6.421) | ||||||

Long-term interest rates | |||||||||

Belgium | 3.834 | 3.521 | 1.176 | 5.676 | -1.778 (4) | -3.261 (4) | 77:QIII | ||

(2.492) | (0.896) | (3.373) | (0.922) | ||||||

Canada | 3.559 | 3.411 | 0.701 | 5.362 | -1.379 (5) | -3.765 (1) | 82:QIV | ||

(2.135) | (0.574) | (1.373) | (1.442) | ||||||

Germany | 4.026 | 44.74 | 2.932 | 4.524 | -2.177 (1) | -3.131 (3) | 82:QIV | ||

(1.154) | (0.841) | (0.842) | (1.183) | ||||||

Japan | 2.490 | 3.720 | -1.937 | 3.903 | -2.032 (4) | -3.474 (3) | 77:QII | ||

(3.812) | (3.034) | (4.628) | (1.002) | ||||||

Netherlands | 2.947 | 1.597 | 1.399 | 5.124 | -1.553 (4) | -3.891 (3) | 77:QII | ||

(2.285) | (1.079) | (2.163) | (1.248) | ||||||

Switzerland | 0.889 | 0.779 | 0.149 | 1.418 | -2.866 (6) | * | -3.964 (6) | 82:QIV | |

(2.658) | (1.242) | (1.768) | (1.184) | ||||||

United Kingdom | 2.303 | 2.746 | -1.732 | 4.233 | -2.178 (6) | -3.577 (4) | 77:QI | ||

(3.265) | (1.489) | (3.916) | (1.650) | ||||||

United States | 3.410 | 2.863 | 0.728 | 5.487 | -2.307 (3) | -5.372 (3) | *** | 80:QI1 | |

(2.237) | (0.386) | (1.327) | (1.640) |

^{1}In percent; standard deviation in parentheses.

^{2}Augmented Die key-Fuller unit root test. Automatic lag length selection according to significance level of last lag of lagged first difference of series. Initial lag length is six lags (lag length in parentheses).

^{3}Minimum t-statistics from a series of augmented Dickey-Fuller regressions with a 1-0 dummy variable in every *period* (excluding observations near end points). Lag lengths in parentheses. Critical values of test statistics taken from Perron and Vogelsang (1992).

^{4}Period in which minimum *t*-statistics occurs.

^{5}Three asterisks denote rejection of the null hypothesis of a unit root in the residuals at the 1 percent level, two asterisks at the 5 percent level, and one asterisk at the 10 percent level.

**Principal Component Analysis of National Short-Term Real Interest Rates, 1960:QI-1994:QIII ^{1}**

^{1}See Appendix III for details.

**Principal Component Analysis of National Short-Term Real Interest Rates, 1960:QI-1994:QIII ^{1}**

United States | United Kingdom | Belgium | Netherlands | Canada | Germany | Switzerland | Japan | |
---|---|---|---|---|---|---|---|---|

Eight countries | ||||||||

Eigenvalues | 3.8953 | 0.9867 | 0.7750 | 0.7132 | 0.6300 | 0.4535 | 0.3043 | 0.2418 |

Eigenvectors | -0.3561 | -0.5841 | 0.0479 | 0.0893 | 0.2498 | -0.1640 | 0.3072 | 0.5815 |

-0.3675 | 0.0492 | 0.1015 | 0.5092 | 0.3640 | 0.6103 | -0.2798 | -0.0987 | |

-0.4071 | 0.0425 | 0.2238 | -0.2008 | -0.4421 | -0.1065 | -0.6605 | 0.3146 | |

-0.3549 | 0.4812 | 0.2297 | 0.0934 | -0.4055 | 0.1733 | 0.6069 | 0.1351 | |

-0.3821 | -0.4008 | 0.3589 | -0.2672 | -0.0545 | -0.0529 | 0.1310 | -0.6858 | |

-0.3198 | 0.5024 | 0.0675 | -0.0810 | 0.5720 | -0.5488 | -0.0462 | -0.0604 | |

-0.3097 | 0.0626 | -0.5904 | -0.6210 | 0.1277 | 0.3807 | 0.0559 | 0.0402 | |

-0.3197 | -0.0785 | -0.6346 | 0.4691 | -0.3143 | -0.3322 | -0.0177 | -0.2437 | |

Percentage of variance | 0.4869 | 0.1233 | 0.0969 | 0.0892 | 0.0788 | 0.0567 | 0.0380 | 0.0302 |

Weight | 0.1268 | 0.1351 | 0.1657 | 0.1260 | 0.1460 | 0.1023 | 0.0959 | 0.1022 |

Three countries | ||||||||

Eigenvalues | 1.6575 | … | … | … | … | 0.7769 | … | 0.5657 |

Eigenvectors | -0.6103 | … | … | … | … | 0.3616 | … | 0.7048 |

-0.5034 | … | … | … | … | -0.8640 | … | 0.0074 | |

-0.6116 | … | … | … | … | 0.3503 | … | -0.7094 | |

Percentage of variance | 0.5525 | … | … | … | … | 0.2590 | … | 0.1886 |

Weight | 0.3725 | … | … | … | … | 0.2534 | … | 0.3741 |

^{1}See Appendix III for details.

**Principal Component Analysis of National Short-Term Real Interest Rates, 1960:QI-1994:QIII ^{1}**

United States | United Kingdom | Belgium | Netherlands | Canada | Germany | Switzerland | Japan | |
---|---|---|---|---|---|---|---|---|

Eight countries | ||||||||

Eigenvalues | 3.8953 | 0.9867 | 0.7750 | 0.7132 | 0.6300 | 0.4535 | 0.3043 | 0.2418 |

Eigenvectors | -0.3561 | -0.5841 | 0.0479 | 0.0893 | 0.2498 | -0.1640 | 0.3072 | 0.5815 |

-0.3675 | 0.0492 | 0.1015 | 0.5092 | 0.3640 | 0.6103 | -0.2798 | -0.0987 | |

-0.4071 | 0.0425 | 0.2238 | -0.2008 | -0.4421 | -0.1065 | -0.6605 | 0.3146 | |

-0.3549 | 0.4812 | 0.2297 | 0.0934 | -0.4055 | 0.1733 | 0.6069 | 0.1351 | |

-0.3821 | -0.4008 | 0.3589 | -0.2672 | -0.0545 | -0.0529 | 0.1310 | -0.6858 | |

-0.3198 | 0.5024 | 0.0675 | -0.0810 | 0.5720 | -0.5488 | -0.0462 | -0.0604 | |

-0.3097 | 0.0626 | -0.5904 | -0.6210 | 0.1277 | 0.3807 | 0.0559 | 0.0402 | |

-0.3197 | -0.0785 | -0.6346 | 0.4691 | -0.3143 | -0.3322 | -0.0177 | -0.2437 | |

Percentage of variance | 0.4869 | 0.1233 | 0.0969 | 0.0892 | 0.0788 | 0.0567 | 0.0380 | 0.0302 |

Weight | 0.1268 | 0.1351 | 0.1657 | 0.1260 | 0.1460 | 0.1023 | 0.0959 | 0.1022 |

Three countries | ||||||||

Eigenvalues | 1.6575 | … | … | … | … | 0.7769 | … | 0.5657 |

Eigenvectors | -0.6103 | … | … | … | … | 0.3616 | … | 0.7048 |

-0.5034 | … | … | … | … | -0.8640 | … | 0.0074 | |

-0.6116 | … | … | … | … | 0.3503 | … | -0.7094 | |

Percentage of variance | 0.5525 | … | … | … | … | 0.2590 | … | 0.1886 |

Weight | 0.3725 | … | … | … | … | 0.2534 | … | 0.3741 |

^{1}See Appendix III for details.

**Principal Component Analysis of National Long-Term Real Interest Rates, 1960:QI-1994:QIII ^{1}**

^{1}See Appendix III for details.

**Principal Component Analysis of National Long-Term Real Interest Rates, 1960:QI-1994:QIII ^{1}**

United Stales | United Kingdom | Belgium | Netherlands | Canada | Germany | Switzerland | Japan | |
---|---|---|---|---|---|---|---|---|

Eight countries | ||||||||

Eigenvalues | 5.1578 | 0.8688 | 0.5982 | 0.4888 | 0.4513 | 0.2321 | 0.1587 | 0.0443 |

Eigenvectors | -0.3395 | -0.5375 | 0.0387 | 0.3936 | -0.1376 | -0.4105 | -0.4082 | 0.2921 |

-0.3591 | -0.3327 | 0.2017 | -0.2730 | -0.3810 | 0.6842 | -0.1430 | -0.1066 | |

-0.4025 | 0.1180 | -0.0070 | -0.1903 | 0.4718 | -0.1801 | 0.1639 | 0.7113 | |

-0.3650 | 0.2597 | 0.3488 | 0.2349 | 0.5227 | 0.0694 | -0.3576 | -0.4653 | |

-0.3944 | -0.2371 | -0.0455 | 0.3098 | 0.0225 | -0.0538 | 0.7792 | -0.2823 | |

-0.3199 | 0.4249 | 0.4859 | -0.2794 | -0.4471 | -0.4326 | 0.1112 | 0.0590 | |

-0.2929 | 0.5211 | -0.5319 | 0.4114 | -0.3541 | 0.2055 | -0.1357 | 0.0668 | |

-0.3418 | -0.1035 | -0.5613 | -0.5795 | 0.1192 | -0.3064 | -0.1421 | -0.3055 | |

Percentage of variance | 0.6447 | 0.1086 | 0.0748 | 0.0611 | 0.0564 | 0.0290 | 0.0198 | 0.0055 |

Weight | 0.1153 | 0.1290 | 0.1620 | 0.1333 | 0.1555 | 0.1024 | 0.0858 | 0.1168 |

Three countries | ||||||||

Eigenvalues | 1.9212 | … | … | … | … | 0.6363 | … | 0.4425 |

Eigenvectors | -0.5851 | … | … | … | … | 0.5146 | … | 0.6268 |

-0.5365 | … | … | … | … | -0.8252 | … | 0.1767 | |

-0.6082 | … | … | … | … | 0.2329 | … | -0.7589 | |

Percentage of variance | 0.6404 | … | … | … | … | 0.2121 | … | 0.1475 |

Weight | 0.3423 | … | … | … | … | 0.2878 | … | 0.3699 |

^{1}See Appendix III for details.

**Principal Component Analysis of National Long-Term Real Interest Rates, 1960:QI-1994:QIII ^{1}**

United Stales | United Kingdom | Belgium | Netherlands | Canada | Germany | Switzerland | Japan | |
---|---|---|---|---|---|---|---|---|

Eight countries | ||||||||

Eigenvalues | 5.1578 | 0.8688 | 0.5982 | 0.4888 | 0.4513 | 0.2321 | 0.1587 | 0.0443 |

Eigenvectors | -0.3395 | -0.5375 | 0.0387 | 0.3936 | -0.1376 | -0.4105 | -0.4082 | 0.2921 |

-0.3591 | -0.3327 | 0.2017 | -0.2730 | -0.3810 | 0.6842 | -0.1430 | -0.1066 | |

-0.4025 | 0.1180 | -0.0070 | -0.1903 | 0.4718 | -0.1801 | 0.1639 | 0.7113 | |

-0.3650 | 0.2597 | 0.3488 | 0.2349 | 0.5227 | 0.0694 | -0.3576 | -0.4653 | |

-0.3944 | -0.2371 | -0.0455 | 0.3098 | 0.0225 | -0.0538 | 0.7792 | -0.2823 | |

-0.3199 | 0.4249 | 0.4859 | -0.2794 | -0.4471 | -0.4326 | 0.1112 | 0.0590 | |

-0.2929 | 0.5211 | -0.5319 | 0.4114 | -0.3541 | 0.2055 | -0.1357 | 0.0668 | |

-0.3418 | -0.1035 | -0.5613 | -0.5795 | 0.1192 | -0.3064 | -0.1421 | -0.3055 | |

Percentage of variance | 0.6447 | 0.1086 | 0.0748 | 0.0611 | 0.0564 | 0.0290 | 0.0198 | 0.0055 |

Weight | 0.1153 | 0.1290 | 0.1620 | 0.1333 | 0.1555 | 0.1024 | 0.0858 | 0.1168 |

Three countries | ||||||||

Eigenvalues | 1.9212 | … | … | … | … | 0.6363 | … | 0.4425 |

Eigenvectors | -0.5851 | … | … | … | … | 0.5146 | … | 0.6268 |

-0.5365 | … | … | … | … | -0.8252 | … | 0.1767 | |

-0.6082 | … | … | … | … | 0.2329 | … | -0.7589 | |

Percentage of variance | 0.6404 | … | … | … | … | 0.2121 | … | 0.1475 |

Weight | 0.3423 | … | … | … | … | 0.2878 | … | 0.3699 |

^{1}See Appendix III for details.

### Common Trend Approach

For the extraction of the common trend in national real interest rates, the cointegration vectors describing the long-run relations between national real inter rates are needed. These were estimated for the period from 1960:QI to 1994:QIII using Johansen’s (1991) maximum likelihood estimator. The lag length of the underlying vector auto regress ion was determined using likelihood ratio tests based initially on eight lags. Two different systems were estimated for both the short- and the long-term real interest rate, including one for the so-called G-3 countries and one for eight countries (see Tables A4 and A5). The outcome was quite sensitive to the number of countries included. The results for the G-3 countries are sensible: two cointegration vectors seem to be significant, so that only one common trend determines the nonstationary part of national real interest rates.

**Cointegration Analysis of Short-Term Real Interest Rates, 1960:QI-1994:QIII**

^{1}Number of cointegration vectors under the null hypothesis (trace statistics).

^{2}Normalized (see Appendix III for details).

**Cointegration Analysis of Short-Term Real Interest Rates, 1960:QI-1994:QIII**

Cointegration Vectors (Row Vectors) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

r^{1} | United States | Germany | Japan | Belgium | Canada | Switzerland | United Kingdom | Netherlands | |||||

1 | 0.1710 | 55.13 | 15.59 | 36.06 | 21.63 | 0.215 | 0.168 | -0.398 | |||||

2 | 0.0636 | 8.80 | 9.52 | 10.93 | 10.47 | -0.537 | 0.275 | 0.142 | |||||

3 | 0.0158 | 2.13 | 2.86 | 2.13 | 2.86 | 0.101 | 0.199 | 0.056 | |||||

βʹ⊥^{2} | 0.8428 | 1.1587 | 0.9457 | ||||||||||

(βʹ⊥β⊥)^{-1}βʹ⊥ | 0.2860 | 0.3931 | 0.3209 | ||||||||||

l | 0.4711 | 81.52 | 44.99 | 274.50 | 135.24 | 0.243 | -0.161 | -0.394 | 0.540 | 0.169 | -0.280 | -0.090 | -0.034 |

2 | 0.3875 | 62.74 | 33.98 | 192.98 | 104.77 | 0.154 | -0.243 | -0.218 | -0.208 | -0.094 | 0.464 | 0.159 | -0.051 |

3 | 0.3405 | 53.29 | 33.62 | 130.24 | 78.36 | -0.261 | 0.090 | -0.015 | -0.077 | -0.236 | -0.130 | 0.095 | 0.300 |

4 | 0.2234 | 32.36 | 27.62 | 76.95 | 55.44 | 0.347 | 0.252 | -0.078 | 0.206 | 0.046 | 0.120 | -0.666 | -0.008 |

5 | 0.1931 | 27.46 | 21.58 | 44.59 | 36.58 | 0.141 | -0.270 | 0.106 | 0.205 | -0.455 | 0.095 | -0.107 | -0.011 |

6 | 0.0730 | 9.70 | 15.59 | 17.14 | 21.63 | 0.377 | -0.199 | -0.006 | -0.143 | 0.117 | 0.075 | -0.029 | 0.024 |

7 | 0.0466 | 6.10 | 9.52 | 7.44 | 10.47 | -0.354 | -0.221 | 0.036 | 0.151 | 0.150 | 0.009 | 0.018 | 0.094 |

8 | 0.0104 | 1.34 | 2.86 | 1.34 | 2.86 | 0.014 | -0.057 | -0.041 | -0.085 | -0.025 | -0.003 | -0.057 | -0.028 |

βʹ⊥^{2} | 0.6481 | 0.8583 | 0.8934 | 1.2912 | 0.3979 | 1.0385 | 1.1586 | 1.0785 | |||||

(βʹ⊥β⊥)^{-1}βʹ⊥ | 0.0880 | 0.1165 | 0.1213 | 0.1753 | 0.0540 | 0.1410 | 0.1573 | 0.1464 |

^{1}Number of cointegration vectors under the null hypothesis (trace statistics).

^{2}Normalized (see Appendix III for details).

**Cointegration Analysis of Short-Term Real Interest Rates, 1960:QI-1994:QIII**

Cointegration Vectors (Row Vectors) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

r^{1} | United States | Germany | Japan | Belgium | Canada | Switzerland | United Kingdom | Netherlands | |||||

1 | 0.1710 | 55.13 | 15.59 | 36.06 | 21.63 | 0.215 | 0.168 | -0.398 | |||||

2 | 0.0636 | 8.80 | 9.52 | 10.93 | 10.47 | -0.537 | 0.275 | 0.142 | |||||

3 | 0.0158 | 2.13 | 2.86 | 2.13 | 2.86 | 0.101 | 0.199 | 0.056 | |||||

βʹ⊥^{2} | 0.8428 | 1.1587 | 0.9457 | ||||||||||

(βʹ⊥β⊥)^{-1}βʹ⊥ | 0.2860 | 0.3931 | 0.3209 | ||||||||||

l | 0.4711 | 81.52 | 44.99 | 274.50 | 135.24 | 0.243 | -0.161 | -0.394 | 0.540 | 0.169 | -0.280 | -0.090 | -0.034 |

2 | 0.3875 | 62.74 | 33.98 | 192.98 | 104.77 | 0.154 | -0.243 | -0.218 | -0.208 | -0.094 | 0.464 | 0.159 | -0.051 |

3 | 0.3405 | 53.29 | 33.62 | 130.24 | 78.36 | -0.261 | 0.090 | -0.015 | -0.077 | -0.236 | -0.130 | 0.095 | 0.300 |

4 | 0.2234 | 32.36 | 27.62 | 76.95 | 55.44 | 0.347 | 0.252 | -0.078 | 0.206 | 0.046 | 0.120 | -0.666 | -0.008 |

5 | 0.1931 | 27.46 | 21.58 | 44.59 | 36.58 | 0.141 | -0.270 | 0.106 | 0.205 | -0.455 | 0.095 | -0.107 | -0.011 |

6 | 0.0730 | 9.70 | 15.59 | 17.14 | 21.63 | 0.377 | -0.199 | -0.006 | -0.143 | 0.117 | 0.075 | -0.029 | 0.024 |

7 | 0.0466 | 6.10 | 9.52 | 7.44 | 10.47 | -0.354 | -0.221 | 0.036 | 0.151 | 0.150 | 0.009 | 0.018 | 0.094 |

8 | 0.0104 | 1.34 | 2.86 | 1.34 | 2.86 | 0.014 | -0.057 | -0.041 | -0.085 | -0.025 | -0.003 | -0.057 | -0.028 |

βʹ⊥^{2} | 0.6481 | 0.8583 | 0.8934 | 1.2912 | 0.3979 | 1.0385 | 1.1586 | 1.0785 | |||||

(βʹ⊥β⊥)^{-1}βʹ⊥ | 0.0880 | 0.1165 | 0.1213 | 0.1753 | 0.0540 | 0.1410 | 0.1573 | 0.1464 |

^{1}Number of cointegration vectors under the null hypothesis (trace statistics).

^{2}Normalized (see Appendix III for details).

**Cointegration Analysis of Long-Term Real Interest Rates, 1960:Q1-1994:QIII**

^{1}Number of cointegration vectors under the null hypothesis (trace statistics).

^{2}Normalized (see Appendix III for details).

**Cointegration Analysis of Long-Term Real Interest Rates, 1960:Q1-1994:QIII**

r^{1} | United States | Germany | Japan | Belgium | Canada | Switzerland | United Kingdom | Netherlands | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.1710 | 55.13 | 15.59 | 36.06 | 21.63 | 0.215 | 0.168 | -0.398 | |||||

0 | 0.1106 | 15.82 | 15.59 | 30.27 | 21.63 | -0.561 | 0.450 | 0.521 | |||||

1 | 0.1000 | 14.23 | 9.52 | 14.45 | 10.47 | 0.040 | 0.152 | -0.320 | |||||

2 | 0.0016 | 0.22 | 2.86 | 0.22 | 2.36 | -0.135 | -0.159 | 0.059 | |||||

βʹ⊥^{2} | 1.0582 | 1.1392 | 0.6727 | ||||||||||

(βʹ⊥β⊥)^{-1}βʹ⊥ | 0.3687 | 0.3969 | 0.2344 | ||||||||||

0 | 0.3408 | 53.33 | 44.99 | 185.11 | 135.24 | 1.106 | 0.151 | 0.239 | 0.719 | -2.023 | -0.204 | -0.050 | -0.108 |

1 | 0.2766 | 41.44 | 38.98 | 131.78 | 104.77 | 0.248 | -0.283 | 0.111 | 0.893 | -0.286 | 1.081 | -0.667 | -0.600 |

2 | 0.2377 | 34.75 | 33.62 | 90.33 | 78.36 | 0.292 | 0.884 | 0.452 | -1.550 | -0.752 | 0.725 | -0.057 | 0.809 |

3 | 0.1677 | 23.50 | 27.62 | 55.59 | 55.44 | 0.054 | -0.082 | 0.529 | -0.632 | 0.818 | -0.448 | -0.495 | -0.017 |

4 | 0.1432 | 19.78 | 21.58 | 32.08 | 36.58 | 1.123 | -0.513 | -0.574 | 1.255 | -0.741 | 0.886 | 0.012 | -1.158 |

5 | 0.0551 | 7.25 | 15.59 | 12.30 | 21.63 | 0.468 | -0.184 | -0.130 | 0.426 | -0.224 | -0.352 | -0.545 | 0.002 |

βʹ⊥^{2} | 0.1512 | 0.1653 | 0.0921 | 0.1653 | 0.1500 | 0.0394 | 0.0945 | 0.1415 | |||||

(βʹ⊥β⊥)^{-1}βʹ⊥ | 1.0860 | 1.1873 | 0.6617 | 1.1873 | 1.0830 | 0.2829 | 0.6787 | 1.0165 |

^{1}Number of cointegration vectors under the null hypothesis (trace statistics).

^{2}Normalized (see Appendix III for details).

**Cointegration Analysis of Long-Term Real Interest Rates, 1960:Q1-1994:QIII**

r^{1} | United States | Germany | Japan | Belgium | Canada | Switzerland | United Kingdom | Netherlands | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.1710 | 55.13 | 15.59 | 36.06 | 21.63 | 0.215 | 0.168 | -0.398 | |||||

0 | 0.1106 | 15.82 | 15.59 | 30.27 | 21.63 | -0.561 | 0.450 | 0.521 | |||||

1 | 0.1000 | 14.23 | 9.52 | 14.45 | 10.47 | 0.040 | 0.152 | -0.320 | |||||

2 | 0.0016 | 0.22 | 2.86 | 0.22 | 2.36 | -0.135 | -0.159 | 0.059 | |||||

βʹ⊥^{2} | 1.0582 | 1.1392 | 0.6727 | ||||||||||

(βʹ⊥β⊥)^{-1}βʹ⊥ | 0.3687 | 0.3969 | 0.2344 | ||||||||||

0 | 0.3408 | 53.33 | 44.99 | 185.11 | 135.24 | 1.106 | 0.151 | 0.239 | 0.719 | -2.023 | -0.204 | -0.050 | -0.108 |

1 | 0.2766 | 41.44 | 38.98 | 131.78 | 104.77 | 0.248 | -0.283 | 0.111 | 0.893 | -0.286 | 1.081 | -0.667 | -0.600 |

2 | 0.2377 | 34.75 | 33.62 | 90.33 | 78.36 | 0.292 | 0.884 | 0.452 | -1.550 | -0.752 | 0.725 | -0.057 | 0.809 |

3 | 0.1677 | 23.50 | 27.62 | 55.59 | 55.44 | 0.054 | -0.082 | 0.529 | -0.632 | 0.818 | -0.448 | -0.495 | -0.017 |

4 | 0.1432 | 19.78 | 21.58 | 32.08 | 36.58 | 1.123 | -0.513 | -0.574 | 1.255 | -0.741 | 0.886 | 0.012 | -1.158 |

5 | 0.0551 | 7.25 | 15.59 | 12.30 | 21.63 | 0.468 | -0.184 | -0.130 | 0.426 | -0.224 | -0.352 | -0.545 | 0.002 |

βʹ⊥^{2} | 0.1512 | 0.1653 | 0.0921 | 0.1653 | 0.1500 | 0.0394 | 0.0945 | 0.1415 | |||||

(βʹ⊥β⊥)^{-1}βʹ⊥ | 1.0860 | 1.1873 | 0.6617 | 1.1873 | 1.0830 | 0.2829 | 0.6787 | 1.0165 |

^{1}Number of cointegration vectors under the null hypothesis (trace statistics).

^{2}Normalized (see Appendix III for details).

In the case of the eight countries, the evidence of one common trend is weaker. Although real interest rates seem to be cointegrated bilaterally (except possibly between Germany and Belgium), the number of significant cointegration vectors in the eight-country estimation system was always less than seven, suggesting more than one common trend. This outcome might be the result of a rapid loss in the power of the multivariate unit root tests caused by the introduction of more countries into the system.^{32} The whole concept of the global real interest rate being associated with a single common trend would therefore be misleading. For this reason, seven cointegration vectors were imposed on a priori reasoning rather than on statistical grounds. This procedure can be justified on the grounds that bilateral real interest rate differentials seem to be stationary in all cases. Nevertheless, given these shortcomings, the conclusions from the common trend calculations should be interpreted with caution.

The estimated cointegration vectors were then used to construct the orthogonal complement β of the matrix of cointegration vectors β, in order to implement Kasa’s (1992) version of the common trend representation for the vector of national real interest rates *X _{t}.^{33}*

Because common trend representations are not unique, some identification scheme needs to be imposed. Kasa’s proposition of normalizing β such that the elements

The results of the cointegration analysis and the weights in the common trend representation are shown in Tables A4 and A5. In the system for the short-term real interest rates of the G-3 countries, the trace statistics suggest two cointegration vectors, whereas the lambda-max statistics suggest one cointegration vector. In the case of long-term real interest rates, both the trace and the lambda-max statistics suggest two cointegration vectors. In the systems for eight countries, there seem to be at most five stationary linear combinations of both short- and long-term real interest rates. It is noteworthy that the weights attached to the countries outside the G-3 group are larger than those inside the group for short-term real interest rates.

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^{}1

This statement ignores the differential effects of taxation, transaction costs, and other measures, which are responsible for incomplete equalization in practice.

^{}2

Theoretically, of course, all rates of return, not just real interest rates, are determined simultaneously in general equilibrium.

^{}3

This seems to be the case according to the most recent empirical evidence. See the reviews by Goldstein and Mussa (1993) and Obstfeld (1993).

^{}4

The standard consumption formula for pricing capital assets for the risk-free real interest rate is given by *y* denotes the coefficient of relative risk aversion (the inverse of the intertemporal elasticity of substitution), *g _{c}* the unconditional mean of the growth rate of consumption (spending), ρ the rate-of-time preference, and

*o*the unconditional variance of the consumption growth rate. This formula is strictly valid only for the unconditional mean of

*n*an inflation risk is not taken into account.

^{}5

Barro and Sala-i-Martin (1990) also included France and Sweden in their analysis, and Barro (1992) included Italy. In terms of country coverage, the sample used in this paper is close to Gagnon and Unferth (1993) and Ford and Laxton (1995), except that Denmark is excluded.

^{}6

The ex ante real short-term interest rate is the more relevant variable from a theoretical perspective, but several studies have shown that short-term ex ante real interest rates are effectively bounded by the short-term horizon and thus cannot deviate from the observed ex post rate by much or for long. See, for example, Barro and Sala-i-Martin (1990) and Laxton, Ricketts, and Rose (1993).

^{}7

This method was chosen in part to avoid the excessive updating generated by estimated recursive univariate models of inflation; the updated coefficient for all countries is 0.3.

^{}8

World real interest rates based on OECD estimates of purchasing power parity exchange rates have also been constructed. The impact on the mean, variance, and unit root properties of the corresponding world real interest rates is so small that it can safely be concluded that the exchange rate choice does not matter for the conclusions drawn in the paper. All calculations are available from the authors upon request.

^{}9

The further results of this analysis confirm that the conclusions do not depend on the interest measure used.

^{}10

The model could be more general; the interest rate could follow an autoregressive process of order *p*, for example. In the case of an AR (1) process, the model would have to rewritten as

^{}11

The main difference between Barro and Sala-i-Manin (1990) and Barro (1992) lies in their aggregation and estimation procedures: in the first article, the world interest rate is modeled and estimated directly, whereas, in the second article, only country-specific interest rates are modeled and estimated.

^{}12

This result is from Barro (1992), who derives results very similar to those in Barro and Sala-i-Martin (1990), except that fiscal variables are not significant in the latter.

^{}15

This very simple device introduces the real interest effects of government debt; see Masson and Knight (1986).

^{}16

In general, ø can be expected to he positive (as in Masson and Knight (1986)). This outcome leaves the exact tax structure in the model unspecified, that is, h^{i} would be income net of taxes. One could also have a ø that is negative and equal to r, as in Diamond (1965). The result in this case would be a fully specified steady state in which the government just rolls over the constant government debt (as a fraction of labor income). It can be shown that the conclusions of this section do not depend on whether ø is positive or negative as long as it lies within the specified range.

^{}18

This representation of the equilibrium is possible if the technology shock in each period is assumed to be distributed log normally (with time-invariant parameters).

^{}21

Stock and Watson (1993) emphasize this point.

^{}22

Given the analogy among world real interest rates, the result; for other interest rates are also analogous. Results with other work real interest rates as left-hand variables are available from the authors upon request.

^{}23

The residuals in the error-correction equations seem to be plagued by heteroscedasticity. For this reason, all standard errors reported are calculated using a heteroscedasticity-consistent variance-covariance matrix of coefficients. The same variance-covariance matrix was used in the process of reducing the general error-correction model to the specific model reported in Table 4.

^{}24

This equation is based on a first-order Taylor expansion around

^{}25

It is, of course, assumed that the endowment shocks for non-tradable goods are less than perfectly correlated across countries.

^{}26

See Stulz (1994) for a discussion of asset pricing in terms of world consumption growth.

^{}27

The global real interest rate as defined in this section is therefore an implication of asset-pricing relations; it does not define an asset-pricing relation. This is a drawback in the sense thai restrictions on global real interest rates are always also restrictions on the asset-pricing model that “prices” the national real interest rates. There is also an advantage to it, however. Assuming that national real interest rates are correctly priced, a global real interest rate can be constructed without specifying the exact underlying asset-pricing model.

^{}29

Univariate fixed and time-varying ARMA (p, q) models—with and without an autoregressive conditional heteroscedasticity (ARCH) specification—were experimented with for the error processes (See Evans (1991) for an AR (1) model with time-varying parameters and an ARCH specification for the forecast errors). In all cases, the estimated models implied a considerable degree of updating to current innovations in forecasts.

^{}30

The (*T* × *k*) matrix of *k* standardized national real interest rates is denoted by *X (T* is the number of observations), and the first eigenvalue and the first eigenvector of *XʹX* with X. and v. The vector of load factors is then given by

^{}31

This result seems not to be specific to the sample period in the calculations. If the principal component analysis is undertaken for the period 1981:QI-1993:QIV, the contribution of the first principal component is remarkably similar to the one for the whole sample period 1960:Q1-1993:QIV. It is striking, however, that the weight for the U.S. real interest rate increases somewhat for both long- and short-term real interest rates.